MinT - Best Known (218−57, 218, s)-Nets in Base 4

Best Known (218−57, 218, s)-Nets in Base 4

(218−57, 218, 531)-Net over F₄ — Constructive and digital

Digital (161, 218, 531)-net over F₄, using

13 times m-reduction [i] based on digital (161, 231, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 77, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(218−57, 218, 576)-Net in Base 4 — Constructive

(161, 218, 576)-net in base 4, using

t-expansion [i] based on (160, 218, 576)-net in base 4, using
- 1 times m-reduction [i] based on (160, 219, 576)-net in base 4, using
  - trace code for nets [i] based on (14, 73, 192)-net in base 64, using
    - 4 times m-reduction [i] based on (14, 77, 192)-net in base 64, using
      - base change [i] based on digital (3, 66, 192)-net over F₁₂₈, using
        net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(218−57, 218, 1625)-Net over F₄ — Digital

Digital (161, 218, 1625)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(218−57, 218, 174498)-Net in Base 4 — Upper bound on s

There is no (161, 218, 174499)-net in base 4, because

1 times m-reduction [i] would yield (161, 217, 174499)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 44365 736279 132082 298374 341425 614410 096819 338979 123561 488107 244768 656644 427066 270563 077167 276144 745222 056217 197078 894842 766423 925672 > 4²¹⁷ [i]